3.1.62 \(\int \frac {1}{1+\cosh ^4(x)} \, dx\) [62]

Optimal. Leaf size=176 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}+2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \coth (x)+2 \coth ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+\sqrt {2} \coth ^2(x)\right ) \]

[Out]

-1/4*arctan((-2*coth(x)+(1+2^(1/2))^(1/2))/(2^(1/2)-1)^(1/2))/(1+2^(1/2))^(1/2)+1/4*arctan((2*coth(x)+(1+2^(1/
2))^(1/2))/(2^(1/2)-1)^(1/2))/(1+2^(1/2))^(1/2)-1/8*ln(2*coth(x)^2+2^(1/2)-2*coth(x)*(1+2^(1/2))^(1/2))*(1+2^(
1/2))^(1/2)+1/8*ln(1+coth(x)^2*2^(1/2)+coth(x)*(2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3288, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\text {ArcTan}\left (\frac {2 \coth (x)+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2} \coth ^2(x)+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^4)^(-1),x]

[Out]

-1/4*ArcTan[(Sqrt[1 + Sqrt[2]] - 2*Coth[x])/Sqrt[-1 + Sqrt[2]]]/Sqrt[1 + Sqrt[2]] + ArcTan[(Sqrt[1 + Sqrt[2]]
+ 2*Coth[x])/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]]) - (Sqrt[1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*
Coth[x] + 2*Coth[x]^2])/8 + (Sqrt[1 + Sqrt[2]]*Log[1 + Sqrt[2*(1 + Sqrt[2])]*Coth[x] + Sqrt[2]*Coth[x]^2])/8

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 3288

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{1+\cosh ^4(x)} \, dx &=\text {Subst}\left (\int \frac {1-x^2}{1-2 x^2+2 x^4} \, dx,x,\coth (x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-\left (1+\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+\left (1+\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ &=\frac {1}{8} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )+\frac {1}{8} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \coth (x)+2 \coth ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+\sqrt {2} \coth ^2(x)\right )-\frac {1}{4} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,-\sqrt {1+\sqrt {2}}+2 \coth (x)\right )-\frac {1}{4} \sqrt {3-2 \sqrt {2}} \text {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {1+\sqrt {2}}+2 \coth (x)\right )\\ &=-\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \coth (x)+2 \coth ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+\sqrt {2} \coth ^2(x)\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 45, normalized size = 0.26 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{2 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{2 \sqrt {1+i}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^4)^(-1),x]

[Out]

ArcTanh[Tanh[x]/Sqrt[1 - I]]/(2*Sqrt[1 - I]) + ArcTanh[Tanh[x]/Sqrt[1 + I]]/(2*Sqrt[1 + I])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.54, size = 37, normalized size = 0.21

method	result	size
risch	\(\munderset {\textit {\_R} =\RootOf \left (512 \textit {\_Z}^{4}-32 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-256 \textit {\_R}^{3}+64 \textit {\_R}^{2}+{\mathrm e}^{2 x}-1\right )\)	\(36\)
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2 \tanh \left (\frac {x}{2}\right ) \textit {\_R} +\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\right )}{4}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cosh(x)^4),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(_R*ln(2*tanh(1/2*x)*_R+tanh(1/2*x)^2+1),_R=RootOf(2*_Z^4-2*_Z^2+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^4),x, algorithm="maxima")

[Out]

integrate(1/(cosh(x)^4 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (124) = 248\).
time = 0.42, size = 590, normalized size = 3.35 \begin {gather*} -\frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left ({\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5\right ) + \frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (-{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{14} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} e^{\left (2 \, x\right )} - \frac {1}{28} \, {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} - {\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} \sqrt {{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5} + \frac {1}{14} \, \sqrt {2} {\left (3 \, \sqrt {2} - 2\right )} - \frac {1}{28} \, {\left ({\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} e^{\left (2 \, x\right )} + 2^{\frac {3}{4}} {\left (2 \, \sqrt {2} + 1\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} - 2\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + \frac {1}{7} \, \sqrt {2} - \frac {3}{7}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{14} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} e^{\left (2 \, x\right )} + \frac {1}{28} \, {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + {\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} \sqrt {-{\left (2^{\frac {3}{4}} e^{\left (2 \, x\right )} + 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} + 4\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} + 4 \, \sqrt {2} + e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 5} - \frac {1}{14} \, \sqrt {2} {\left (3 \, \sqrt {2} - 2\right )} - \frac {1}{28} \, {\left ({\left (2^{\frac {3}{4}} {\left (8 \, \sqrt {2} + 11\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (5 \, \sqrt {2} + 6\right )}\right )} e^{\left (2 \, x\right )} + 2^{\frac {3}{4}} {\left (2 \, \sqrt {2} + 1\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, \sqrt {2} - 2\right )}\right )} \sqrt {-2 \, \sqrt {2} + 4} - \frac {1}{7} \, \sqrt {2} + \frac {3}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^4),x, algorithm="fricas")

[Out]

-1/16*2^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4)*log((2^(3/4)*e^(2*x) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(
2) + 4) + 4*sqrt(2) + e^(4*x) + 2*e^(2*x) + 5) + 1/16*2^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4)*log(-(2^(3/4)
*e^(2*x) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + e^(4*x) + 2*e^(2*x) + 5) + 1/4*2^(1/4)*
sqrt(-2*sqrt(2) + 4)*arctan(1/14*(sqrt(2)*(5*sqrt(2) + 6) + 8*sqrt(2) + 4)*e^(2*x) - 1/28*(2*sqrt(2)*(5*sqrt(2
) + 6) - (2^(3/4)*(8*sqrt(2) + 11) + 2*2^(1/4)*(5*sqrt(2) + 6))*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 8)*sqrt((2
^(3/4)*e^(2*x) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + e^(4*x) + 2*e^(2*x) + 5) + 1/14*s
qrt(2)*(3*sqrt(2) - 2) - 1/28*((2^(3/4)*(8*sqrt(2) + 11) + 2*2^(1/4)*(5*sqrt(2) + 6))*e^(2*x) + 2^(3/4)*(2*sqr
t(2) + 1) + 2*2^(1/4)*(3*sqrt(2) - 2))*sqrt(-2*sqrt(2) + 4) + 1/7*sqrt(2) - 3/7) + 1/4*2^(1/4)*sqrt(-2*sqrt(2)
+ 4)*arctan(-1/14*(sqrt(2)*(5*sqrt(2) + 6) + 8*sqrt(2) + 4)*e^(2*x) + 1/28*(2*sqrt(2)*(5*sqrt(2) + 6) + (2^(3
/4)*(8*sqrt(2) + 11) + 2*2^(1/4)*(5*sqrt(2) + 6))*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 8)*sqrt(-(2^(3/4)*e^(2*x
) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + e^(4*x) + 2*e^(2*x) + 5) - 1/14*sqrt(2)*(3*sqr
t(2) - 2) - 1/28*((2^(3/4)*(8*sqrt(2) + 11) + 2*2^(1/4)*(5*sqrt(2) + 6))*e^(2*x) + 2^(3/4)*(2*sqrt(2) + 1) + 2
*2^(1/4)*(3*sqrt(2) - 2))*sqrt(-2*sqrt(2) + 4) - 1/7*sqrt(2) + 3/7)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)**4),x)

[Out]

Timed out

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Giac [C] Result contains complex when optimal does not.
time = 0.46, size = 281, normalized size = 1.60 \begin {gather*} -\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} + 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} + 50 \, \sqrt {2} - \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} + 70\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} - 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} + 50 \, \sqrt {2} + \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} + 70\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} + 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + \left (4 i + 2\right ) \, \sqrt {2} + \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} - 4 i - 2\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} - 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + \left (4 i + 2\right ) \, \sqrt {2} - \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} - 4 i - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^4),x, algorithm="giac")

[Out]

-(1/16*I + 1/16)*sqrt(2*sqrt(2) - 2)*(-I/(sqrt(2) - 1) + 1)*log((20*I + 10)*sqrt(2)*e^(2*x) + 10*sqrt(2)*sqrt(
10*sqrt(2) + 14) + 50*sqrt(2) - (2*I - 14)*sqrt(10*sqrt(2) + 14) + (28*I + 14)*e^(2*x) + 70) + (1/16*I + 1/16)
*sqrt(2*sqrt(2) - 2)*(-I/(sqrt(2) - 1) + 1)*log((20*I + 10)*sqrt(2)*e^(2*x) - 10*sqrt(2)*sqrt(10*sqrt(2) + 14)
+ 50*sqrt(2) + (2*I - 14)*sqrt(10*sqrt(2) + 14) + (28*I + 14)*e^(2*x) + 70) - (1/16*I + 1/16)*sqrt(2*sqrt(2)
+ 2)*(-I/(sqrt(2) + 1) + 1)*log(2*sqrt(2)*e^(2*x) + 2*sqrt(2)*sqrt(2*sqrt(2) - 2) + (4*I + 2)*sqrt(2) + (2*I -
2)*sqrt(2*sqrt(2) - 2) - 2*e^(2*x) - 4*I - 2) + (1/16*I + 1/16)*sqrt(2*sqrt(2) + 2)*(-I/(sqrt(2) + 1) + 1)*lo
g(2*sqrt(2)*e^(2*x) - 2*sqrt(2)*sqrt(2*sqrt(2) - 2) + (4*I + 2)*sqrt(2) - (2*I - 2)*sqrt(2*sqrt(2) - 2) - 2*e^
(2*x) - 4*I - 2)

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Mupad [B]
time = 1.01, size = 205, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240+149422080{}\mathrm {i}\right )[phone]+94306304{}\mathrm {i}\right )}{8}-\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240-149422080{}\mathrm {i}\right )[phone]+94306304{}\mathrm {i}\right )}{8}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240-149422080{}\mathrm {i}\right )[phone]-94306304{}\mathrm {i}\right )}{8}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240+149422080{}\mathrm {i}\right )[phone]-94306304{}\mathrm {i}\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^4 + 1),x)

[Out]

(2^(1/2)*(1 - 1i)^(1/2)*log(exp(2*x)*(436273152 + 91291648i) - 2^(1/2)*(1 - 1i)^(1/2)*(9830400 - 56623104i) +
2^(1/2)*(1 - 1i)^(1/2)*exp(2*x)*(218890240 + 149422080i) + (21168128 + 94306304i)))/8 - (2^(1/2)*(1 - 1i)^(1/2
)*log(exp(2*x)*(436273152 + 91291648i) + 2^(1/2)*(1 - 1i)^(1/2)*(9830400 - 56623104i) - 2^(1/2)*(1 - 1i)^(1/2)
*exp(2*x)*(218890240 + 149422080i) + (21168128 + 94306304i)))/8 + (2^(1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(436273
152 - 91291648i) - 2^(1/2)*(1 + 1i)^(1/2)*(9830400 + 56623104i) + 2^(1/2)*(1 + 1i)^(1/2)*exp(2*x)*(218890240 -
149422080i) + (21168128 - 94306304i)))/8 - (2^(1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(436273152 - 91291648i) + 2^(
1/2)*(1 + 1i)^(1/2)*(9830400 + 56623104i) - 2^(1/2)*(1 + 1i)^(1/2)*exp(2*x)*(218890240 - 149422080i) + (211681
28 - 94306304i)))/8

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